Question

Two discs A and B of same material and thickness have radii \(R\) and \(3R\) respectively. Their moments of inertia about their axis will be in the ratio

• A. \(3 : 1\)
• B. \(1 : 9\)
• C. \(1 : 81\)
• D. \(1 : 27\)

Hint:

For discs of same material/thickness: Mass $\propto R^2$. Since $I$ depends on $MR^2$, the final dependence is $R^4$.

Answer 1

The Correct Option is C

Step 1: Concept
Moment of inertia of a disc is $I = \frac{1}{2}MR^2$. Since they have the same material and thickness ($t$), mass $M = \rho \times V = \rho \times (\pi R^2 t)$.

Step 2: Meaning
Thus, $M \propto R^2$. Substituting this into the $I$ formula: $I \propto (R^2) \times R^2 \implies I \propto R^4$.

Step 3: Analysis
$\frac{I_A}{I_B} = \left( \frac{R_A}{R_B} \right)^4$
$\frac{I_A}{I_B} = \left( \frac{R}{3R} \right)^4 = \left( \frac{1}{3} \right)^4 = \frac{1}{81}$.

Step 4: Conclusion
The ratio of moments of inertia is $1 : 81$. Final Answer: (C)